My young friend, any of the cartridges you mention will do a fine job on medium game. All are great target rounds, and you'll never ever regret owning a rifle chambered in any of them. However, if you don't handload you may find ammunition for the .270 and .30-06 to be more readily available and probably cheaper than the 7mm-08.
Have you actually fired a .270 or .30-06 to see how bad the recoil really is? You might find that you get used to it. The recoil difference from a 7mm-08 is certainly MEASUREABLE, but may or may not be NOTICEABLE on your shoulder.
Besides a good recoil pad on the rifle, you should also get a shoulder pad. This will help tremendously in avoiding pain, suffering and damage to your rotator cuff that will require surgery. It will alos help you concentrate on shooting. PAST makes these for about $20 to $30 depending on the thickness. I use one at all times when at the range or shooting clay birds and will never be without one. Mine is just the low-end Field Grade. Here's a link to Midway that describes some of them:
http://www.midwayusa.com/rewriteabro...0/655***688***